Nets Katz, Terence Tao
The purpose of this article is to survey the developments on the Kakeya problem in recent years, concentrating on the period after the excellent 1999 survey of Wolff, and including some recent work by the authors. We will focus on the standard Kakeya problem for line segments and not discuss other important variants (such as Kakeya estimates for circles, light rays, or $k$-planes).
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Equivalence of linear and trilinear Kakeya conjectures in three dimensions
本文证明三维中 Kakeya 极大算子猜想与其不相交三线性对偶形式彼此等价。
Sticky Kakeya sets and the sticky Kakeya conjecture
本文提出并证明了粘性Kakeya集的猜想，表明其Hausdorff和Minkowski维度为n。
Bennett, Carbery, and Tao formulated an n-linear analogue of the Kakeya conjecture in R^n. They proved the conjecture except for the endpoint case. We prove the endpoint case.
Here we show some results related with Kakeya conjecture which says that for any integer $n\geq 2$, a set containing line segments in every dimension in $\mathbb{R}^n$ has full Hausdorff dimension as well as box dimension. We proved here that the Kakeya books, which are Kakeya sets with some restrictions on positions of line segments have full box dimension. We also prove here a relation between the projection property of Kakeya sets and the Kakeya conjecture. If for any Kakeya set $K\subset\mathbb{R}^n$, the Hausdorff dimension of orthogonal projections on $k\leq n$ subspaces is independent o…
The Kakeya conjecture, after Wang and Zahl
本文综述了Wang和Zahl证明三维Kakeya猜想的证明思路，通过图表展示核心方法与主要贡献。
Rotating needles in space: the road to the Kakeya conjecture, and why it matters
本文对卡克亚猜想作非技术性阐释，说明其重要性，介绍Hong Wang与Joshua Zahl在三维空间中解决该猜想的研究历程。
Introduction to the proof of the Kakeya conjecture
本文综述 Hong Wang 与 Joshua Zahl 对三维 Kakeya 猜想的证明，梳理问题背景、既有工作、核心困难与新方法。
A streamlined proof of the Kakeya set conjecture in $\mathbb{R}^3$
本文提供了实数三维空间中Kakeya集猜想的简洁证明，简化了原有证明过程。
We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in $\R^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called "joints" problem, as well as presenting some $n$-linear analogues for $n<d$.
A Survey of the Kakeya conjecture, 2000-2025
本文综述了2000-2025年间Kakeya猜想在欧几里得空间中的研究进展，重点介绍自Wolff和Katz-Tao之后的新发展。
Through the use of a nonstandard version of Frostman's lemma, the notion of Hausdorff dimension is formulated in nonstandard euclidean space of arbitrary dimension. This allows for a nonstandard proof of the Kakeya conjecture in two dimensions, which can then be extended to the three-dimensional case.
Outline of the Wang-Zahl proof of the Kakeya conjecture in $\mathbb{R}^3$
本文概述了Wang-Zahl关于三维Kakeya猜想由粘性情形推出的证明，该证明基于Katz-Tao的方法。
Lifting curved Kakeya sets to linear Kakeya sets
本文证明弯曲Kakeya问题可提升为高维线性Kakeya问题，若经典猜想成立，则全维曲线族在Hörmander型族中稠密，并深化了对相关问题的理解。
A comparison of trilinear testing conditions for the paraboloid Fourier extension and Kakeya conjectures in three dimensions
本文比较了抛物面傅里叶扩展猜想与卡凯ya猜想的三重线性测试条件，将调制测试条件转换为受限的光滑Alpert测试条件。
Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the $(L^p,L^p)$ spherical restriction theorem of Wolff from $p > 42/11$ to $p > 34/9$, and also obtain a sharp $(L^p,L^q)$ spherical restriction theorem for $q > 4 - 5/27$.
A Besicovitch set is a subset of $\R^d$ that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this conjecture in a variety of contexts. Our proofs are simple and aim to give an intuitive feel for the problem. For example, we give a very simple proof that the packing and lower box-counting dimension of any Besicovitch set is at least $(d+1)/2$ (better estimates are available in the literature). We also study the `generic validity' of the Kakeya conjecture in the …
We state and prove an analogue of the Kakeya conjecture for the local field $\mathbb{F}_q(\!(t)\!)$. This extends Arsovski's result on the Kakeya conjecture to local fields of positive characteristic. We also prove the Kakeya maximal conjecture in this setting.
An improved bound on the Hausdorff dimension of sticky Kakeya sets in $\mathbb{R}^4$
本文针对$\u211d^4$中的粘性Kakeya集，结合平面刷论证与粘性性质，将其Hausdorff维数下界从原有3.059提升至3.25，推进了粘性Kakeya猜想的相关研究。
The Kakeya Conjecture: where does it come from and why is it important?
本文探讨了Kakeya猜想的起源及其在傅里叶分析中的重要性，并介绍了Hong Wang和Joshua Zahl的突破性证明。
We prove that the Kakeya maximal conjecture is equivalent to the $Ω$-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{é} proved that the Kakeya conjecture is equivalent to the $Ω$-Kakeya conjecture. Moreover, we improve concrete bound on the Hausdorff dimension of a $Ω$-Kakeya set : for any Bore set $Ω$ in S n--1 , we prove that if X $\subset$ R n contains for any e $\in$ $Ω$ a unit segment oriented along e then we have dX $\ge$ 6 11 d$Ω$ + 1 where dE denotes the Hausdorff dimension of a set E.
We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.
We prove that all bounded subsets of $\mathbb{Q}_p^n$ containing a line segment of unit length in every direction have Hausdorff and Minkowski dimension $n$. This is the analogue of the classical Kakeya conjecture with $\mathbb{R}$ replaced by $\mathbb{Q}_p$.
We obtain new bounds for the Kakeya maximal conjecture in most dimensions $n<100$, as well as improved bounds for the Kakeya set conjecture when $n=7$ or $9$. For this we consider Guth and Zahl's strengthened formulation of the maximal conjecture, concerning families of tubes that satisfy the polynomial Wolff axioms. Our results give improved estimates for this strengthened formulation when $n =5$ or $n \geq 7$.
The arithmetic Kakeya conjecture, formulated by Katz and Tao in 2002, is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in $\mathbb{R}^n$ is $n$. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds.
We study a variety of problems about homothets of sets related to the Kakeya conjecture. In particular, we show many of these problems are equivalent to the arithmetic Kakeya conjecture of Katz and Tao. We also provide a proof that the arithmetic Kakeya conjecture implies the Kakeya conjecture for packing dimension, as this implication was previously only known for Minkowski dimension. We consider several questions analogous to the classical results of Stein and Bourgain about the Lebesgue measure of a set containing a sphere centered at every point of $[0,1]^n$, where we replace spheres by ar…
A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least $C_{n,ε} N^{n - ε}$ for any $ε$ -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime $N$. We also show that the case of general $N$ can be reduced to lower bounding the $\mathbb{F}_p$ rank of the incidence matrix of points and hyperplanes over $(\mathbb{Z}/p^k\mathbb{Z})^n$.
Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions
本文研究三维空间中具有限制性管集合的并集体积，证明其体积接近最大，并由此推出所有Kakeya集的Minkowski和Hausdorff维数均为3。
Bourgain's condition, sticky Kakeya, and new examples
本文证明在三维及以上维度，满足Bourgain条件的Hörmander型振荡积分算子的粘性Kakeya猜想可归约至经典Kakeya猜想，支持Guo-Wang-Zhang的猜想，同时提供新例子展示该性质不随管径增大而保持。
Kakeya and Furstenberg problems for sticky sets of tubes
本文研究粘性管集的关联问题，证明了三维空间中满足凸Wolff公理的粘性管集的尖锐Furstenberg集估计，以及四维空间中相应的Kakeya集猜想。
Let $A, B$, be finite subsets of an abelian group, and let $G \subset A \times B$ be such that $# A, # B, # \{a+b: (a,b) \in G \} \leq N$. We consider the question of estimating the quantity $# \{a-b: (a,b) \in G \}$. Recently Bourgain improved the trivial upper bound of $N^2$ to $N^{2-1/13}$, and applied this to the Kakeya conjecture. We improve Bourgain's estimate further to $N^{2-1/6}$, and obtain the further improvement of $N^{2-1/4}$ if we also know that $# \{a+2b: (a,b) \in G\} \leq N$. We conclude that Besicovitch sets in $\R^n$ have Hausdorff dimension at least 6n/11+5/11 and Minkowski…
In the finite field setting, we show that the restriction conjecture associated to any one of a large family of $d=2n+1$ dimensional quadratic surfaces implies the $n+1$ dimensional Kakeya conjecture (Dvir's theorem). This includes the case of the paraboloid over finite fields in which $-1$ \emph{is} a square. We are able to partially reverse this implication using the sharp Kakeya maximal operator estimates of Ellenberg, Oberlin and Tao to establish the first finite field restriction estimates beyond the Stein-Tomas exponent in this setting.
We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization which completely solves the discrete analogue of the Furstenberg set problem in all dimensions. The difference between our theorems and the (true) problems is only the (still difficult) issue of continuity since no transversality-at-incidences assumptions are imposed. The main tool of the proof is a theorem of Wongkew \cite{wongkew2003volumes} which states that…
We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb R}^3$, there are at most $O(N^{3 \over 2})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines in ${\Bbb R}^3$ so that no $N$ lines lie in the same plane and so that each line intersects a set $P$ of points in at least $N$ points then the cardinality of the set of points is $Ω(N^3)$. Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.
Kakeya conjecture and High-Rank Lattice von Neumann algebras
本文探讨了非交换Lp空间的完全有界近似性质与Kakeya猜想之间的关系，通过球面分析方法和Zygmund光滑性条件，推导出相关结论。
The Kakeya problem for codimension-two planes in $\mathbb{R}^n$
研究n≥4时ℝⁿ中满足凸Wolff公理的余维二平面Kakeya问题，证明此类δ片的并集具有几乎最大体积，是Wang和Zahl工作的高维推广，还解决了ℝ⁴中满足曲率条件的Rogers 2平面Kakeya问题。
Curved Kakeya sets and Nikodym problems on manifolds
本文将满足 Bourgain 条件的弯曲 Kakeya 集及常曲率流形上的 Nikodym、Furstenberg 问题归结为欧氏对应问题，从而证明三维常截面曲率流形的 Nikodym 猜想和二维常 Gauss 曲率曲面的 Furstenberg 猜想。
Kakeya Conjecture and Conditional Kolmogorov Complexity
本文提出一个信息论框架，探讨在纤维化几何集中标注信息对残余描述长度的压缩影响，通过算法维度和Lutz点到集原理，证明点的复杂性可分解为纤维标签和沿纤维残余复杂性之和，解决Kakeya猜想在高维空间中的挑战。
Refinements on the Restriction and Kakeya problems range in $\mathbb{R}^4$
通过证明两个两端点关联估计，改进了 $\mathbb{R}^4$ 中 Fourier 限制性与 Kakeya 极大估计的范围，并探讨了对 Kakeya 集维数下界和 Bochner--Riesz 猜想的影响。
Kakeya sets and dimension compression in compact Lie groups
本文针对紧半单李群中的两类Kakeya集问题，确定了其最优Minkowski维数的上下界，推测局部Kakeya集问题的上界为精确值，结合李理论构造、关联几何与数的几何完成相关证明。
This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in $\mathbb{R}^3$, what we expect to hold due to the maximal Kakeya operator conjecture, as well as…
Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements
本文证明 Kakeya 猜想在紧/Borel/任意集、线段/直线及大方向集等表述下等价，并通过加性补集建立 Hausdorff 维数与 packing 维数的对偶公式。
The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and other hypersurfaces. As the field is quite large and has so many applications, it will be impossible to completely survey the field, but we will try to at least give the main ideas and developments in this area. The restriction problem are connected to many other conjectures, notably the Kakeya and Bochner-Riesz conjectures, as well as PDE conjectures such as t…
We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.
We survey the interconnections between geometric combinatorics (such as the Kakeya problem), arithmetic combinatorics (such as the classical problem of determining which sets contain arithmetic progressions), oscillatory integrals (such as the Bochner-Riesz, restriction, and local smoothing problems), and the local and global well-posedness theory for non-linear dispersive and wave equations.
We survey recent developments on the Restriction conjecture.
In this article, we collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. We start with a sketch of Milne's proof of two of these conjectures. We also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of $4s^2$, respectively $4s(s+1)$, triangular numbers. This approach is inspired by Zagier's proof of the Kac--Wakimoto formulas via modular forms. We end the survey with recent conjectures of the first author and Chua.
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